Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

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A kaleidoscopic coloring of the plane

Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\...
Taras Banakh's user avatar
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24 votes
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Example of a quasi-Bernoulli measure which is not Gibbs?

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider measures on $X$ only. A measure $\mu$ is quasi-Bernoulli if there is a constant $C\ge 1$ such that for any finite sequences $i,j$, $$ C^{-1} \...
Pablo Shmerkin's user avatar
23 votes
0 answers
921 views

A question about small sets of reals

In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, we have that $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$? This question is ...
Ashutosh's user avatar
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21 votes
0 answers
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A question about sigma algebras and rectangles

Let P be the statement: Every subset of plane belongs to the sigma algebra generated by $\{A \times B : A, B \subseteq \mathbb{R}\}$. Let Q be the statement: Every continuum-sized family of subsets ...
Ashutosh's user avatar
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18 votes
1 answer
2k views

Function of two sets intersection

Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
pi66's user avatar
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16 votes
0 answers
246 views

Gap two Sierpinski set?

Is it consistent to have a set of reals $X$ of size $\aleph_3$ such that for every $Y \subseteq X$, $Y$ has measure zero iff $|Y| \leq \aleph_1$?
Ashutosh's user avatar
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15 votes
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1k views

A Kakeya-like problem: must a union of annuli fill the plane?

Let $S$ be a subset of $\mathbb{R}^2$ with the following property. For all $x \in \mathbb{R}^2$ and $\varepsilon \gt 0$, there exists a nontrivial interval $[a,b] \subseteq [1-\varepsilon,1]$, such ...
Scott Aaronson's user avatar
14 votes
0 answers
481 views

Lebesgue density 1/2 (or bounded away from 0 and 1)

From the work of Preiss, we know that in infinite-dimensional spaces, one has violations of the Lebesgue density theorem. In particular, he has constructed examples of probability spaces where a set ...
Aryeh Kontorovich's user avatar
14 votes
0 answers
418 views

A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?

Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...
Taras Banakh's user avatar
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12 votes
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Weak$^*$ convergence of measures vs. convergence of supports

Let $X$ be a compact metric space and let $\mathcal M(X)$ denote the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\text{supp} \mu$ for the support of $\mu$. It is easy to ...
Dominik Kwietniak's user avatar
12 votes
0 answers
428 views

Uniform closure of subspaces of Baire class 1

Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...
Fred Dashiell's user avatar
11 votes
0 answers
479 views

Are there 100 points that are part of every half-density part of the plane?

Is there a configuration $P$ that consists of 100 points of the plane such that every $X\subset\mathbb R^2$ whose density is half contains an isometric copy of $P$? I am deliberately being vague ...
domotorp's user avatar
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11 votes
0 answers
246 views

Which results in probabilistic group theory generalize from finite groups to compact Hausdorff groups (and which don't)?

Let $G$ be a finite group. It has been shown that: If the probability that two randomly selected elements of $G$ generate an abelian group is greater than $5/8$, $G$ is abelian. If the probability ...
ckefa's user avatar
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11 votes
0 answers
212 views

Shift invariant measurable selection theorem

Let $(X,\mathcal{F})$ be some measure space and endow $\mathbb{R}^\mathbb{Z}$ with the product topology and borel $\sigma$-field. Let $F$ be a point to set mapping $X^\mathbb{Z}\rightarrow \mathcal{P}(...
Marc's user avatar
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11 votes
0 answers
779 views

Errata for the Treatise of Analysis of Dieudonné

I was looking again at the beautiful and quite complete work of Dieudonné, his Treatise of Analysis, to refresh my memory about some aspects of classical analysis. I especially love this Treatise for ...
brunoh's user avatar
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11 votes
0 answers
3k views

Quotients of Measurable Spaces?

Let $(\Omega,\Sigma)$ be a measurable space and $\Pi$ be a partition of $\Omega$. There is a projection $\pi:\Omega\to\Pi$ that maps each $\omega\in\Omega$ to the unique partition cell in $\Pi$ ...
Michael Greinecker's user avatar
10 votes
0 answers
254 views

What is the smallest $\sigma$-algebra of reals that is closed under addition of sets?

What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$ I know that neither ...
Alessandro Codenotti's user avatar
10 votes
0 answers
367 views

Concerning Luzin-(N)-property

Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set. By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
喻 良's user avatar
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10 votes
0 answers
164 views

Maximizing an integral w.r.t. a measure on the unit sphere

I would like to know if the answer to the following question is known. Let $d \ge 3$. What is the value of $$ \theta(d) := \max_{\mu} \int_{S^{d-1}} \int_{S^{d-1}} \cdots \int_{S^{d-1}} |x_1 \...
Romeo's user avatar
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10 votes
0 answers
467 views

A subset of plane that meets every line at length one

Is there a subset of plane whose intersection with every line has length one? It is easy to construct such a set under the continuum hypothesis. Also, no such set is Lebesgue measurable - See ...
Ashutosh's user avatar
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10 votes
0 answers
725 views

Is the set of real-valued lower semi-continuous functions measurable in epigraph topology (= topology of Gamma convergence)?

Let LSC = LSC([0,1]) be the set of non-negative, lower semi-continuous functions on the unit interval which take values in $\mathbb{R}_+ \cup \{\infty\}$. We use epigraph topology on LSC, i.e. a ...
Wolfgang Loehr's user avatar
9 votes
0 answers
244 views

Is the inverse of a measurably parametrised family of bijections between standard Borel spaces measurably parametrised?

It is known that a measurable bijection $f \colon [0,1] \to [0,1]$ has a measurable inverse. (Here, all measurability is simply with respect to the Borel $\sigma$-algebra of $[0,1]$.) Now fix an ...
Julian Newman's user avatar
9 votes
0 answers
216 views

Point-free topology, but with $\sigma$-algebras instead of spaces

I have a question about $\sigma$-algebras in relation to point-free topology. The question was inspired by a comment on a similar question I had: If abstract $\sigma$-algebras (i.e. certain boolean ...
Ronald J. Zallman's user avatar
9 votes
0 answers
246 views

Covering inequality for sets of intervals

Let $I$ and $J$ be finite sets of open intervals $(a,b)\subset\mathbb R$. For a finite set of points $P\subset \mathbb R$ we denote those subsets of intervals from $I$ and $J$ containing some point ...
Julian's user avatar
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9 votes
0 answers
597 views

Two questions about universally measurable sets

I have two questions about universally measurable sets: (1) Is there a universally measurable set of reals which does not have the Baire property? (2) Is there a universally measurable set of reals ...
Ashutosh's user avatar
  • 9,771
9 votes
0 answers
219 views

Cramer's theorem in Hilbert spaces

I am interested under what conditions Cramer's theorem applies in random variables taking values in Hilbert spaces. Following these lecture notes, but using a Hilbert space: Let $X_1,X_2,\cdots$, be ...
Manuel Schmidt's user avatar
9 votes
0 answers
190 views

A small planar set containing a large family of curves

A beautiful construction by Besicovitch and Rado [1] produces an astounding example of a compact connected plane set of measure zero containing circles of all radii $r\in(0,1]$. A corollary to a ...
Wlodek Kuperberg's user avatar
8 votes
0 answers
379 views

Non-affine smooth transformation of Gaussian is Gaussian

Suppose $Z\sim N(0,1)$ (standard Gaussian) and $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(Z)\sim N(0,1)$. My question is whether there exists any such $f$ other than $f(x)...
De vinci's user avatar
  • 329
8 votes
0 answers
304 views

The Cauchy Transform, and the convergence of the Fourier-Stieltjes transforms of a sequence of measures

Let $C\left(\mathbb{R}/\mathbb{Z}\right)$ denote the Banach space of continuous, $1$-periodic complex-valued functions on the unit interval, let $M\left(\mathbb{R}/\mathbb{Z}\right)$ denote its dual, ...
MCS's user avatar
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8 votes
0 answers
1k views

Can every set be measurable?

The Solovay model shows that ZF (plus an inaccessible) is consistent with every subset of $\mathbb{R}$ being measurable. How far can we go in that direction? We can always have non-measurable sets ...
arsmath's user avatar
  • 6,720
8 votes
0 answers
435 views

When is the sigma-algebra generated by closed convex sets the same as the Borel sigma-algebra

For which topological vector spaces $E$ do we have the equality between the sigma algebra generated by the closed convex subsets of $E$ and the Borel sigma algebra of $E$ ? More precisely, do we have ...
LCO's user avatar
  • 496
8 votes
0 answers
528 views

A Banach-Tarski game

This is partially inspired by the question https://math.stackexchange.com/questions/1383397/cutting-a-banach-tarski-cake, which I find intriguing if unclearly written. A paradoxical family of subsets ...
Noah Schweber's user avatar
8 votes
0 answers
210 views

Qualitative weakenings of probabilistic independence

In probability theory, independence of random variables is characterised by $$(1)~~X~\text{independent}~Y \; \iff \; P_{(X,Y)} = P_X \otimes P_Y \enspace ,$$ where $P_{(X,Y)}$ is the joint probability ...
Alex Simpson's user avatar
8 votes
0 answers
698 views

Density of countably additive measure in the set of all finitely additive measures.

Let $S$ be a countable discrete set, the following two results are quite easy to prove: Every countably additive probability measure $\mu$ on $S$ commutes (in Fubini's sense) with every finitely ...
Valerio Capraro's user avatar
8 votes
0 answers
1k views

G-delta of measure 0 containig the rationals.

It is well known that $\mathbb{Q}$ is not a $G_\delta$. In fact no countable dense subset of $\mathbb{R}$ is a $G_\delta$. We order the rationals in a sequence: $\mathbb{Q}=\lbrace r_k\rbrace$, and ...
Julián Aguirre's user avatar
8 votes
1 answer
191 views

Subspaces of $L_p([0,1])$ whose unit ball is compact for the topology of convergence in measure

Any information about the following questions would be welcome. I wonder whether there are (well-known or easy) closed and infinite dimensional subspaces of $L_p([0,1])$ ($1<p<\infty$) whose ...
user159631's user avatar
7 votes
0 answers
128 views

Relation between the additive Haar measure on $(K,+)$ and the multiplicative Haar measure on $K^{*}$ for a global field $K$

The following question comes from my studying of Alain Connes's paper Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function. In it, on p. 11, Connes notes that if $K$ is ...
The Thin Whistler's user avatar
7 votes
0 answers
154 views

Relationship between measure theory and quantification

I was advised that this question might be better suited for mathoverflow, so I am reposting it here (original post). In a 1978 paper published by David P. Ellerman and Gian-Carlo Rota, the duo discuss ...
John Smith's user avatar
7 votes
0 answers
152 views

Divergence as infinitesimal volume change on a Finsler manifold

Let $M$ be a smooth manifold and $Z$ a smooth vector field on it. It generates a family of diffeomorphisms $\phi_t:M\to M$ by demanding that $\phi_0=\operatorname{id}$ and $\partial_t\phi_t(x)=Z(\...
Joonas Ilmavirta's user avatar
7 votes
0 answers
254 views

Composition of couplings as a pullback construction

A metric measure space $(X,d,\mu)$ consists of a metric space $(X,d)$ together with a Borel measure $\mu$. A coupling between metric measure spaces $(X_1,d_1,\mu_1)$ and $(X_2,d_2,\mu_2)$ consists of ...
george's user avatar
  • 554
7 votes
0 answers
176 views

Does this ideal in $B(L_1)$ have a (bounded) right approximate identity?

I will take a roundabout way to defining this ideal, because (a) this route is how my collaborators and I came to it (b) this alternative definition, rather than the standard one, may suggest a ...
Yemon Choi's user avatar
  • 25.5k
7 votes
0 answers
420 views

A discontinuous construction

Suppose we have an uncountable family of functions $f_r: [0, 1] \to R$ indexed by $r \in [0, 1]$ such that for each $r$, there exists a unique $x$ in $[0, 1]$ such that $f_{r}$ is positive on $x$ and $...
James Baxter's user avatar
  • 2,029
7 votes
0 answers
95 views

Volume of a neighborhood of singular matrices

Suppose we take the set of all $n\times n$ real matrices with entries in $[0,1]$ in Euclidean space. Let $N_\epsilon$ be the $\epsilon$ neighborhood of the set of all singular matrices in this space, ...
zumbaya's user avatar
  • 71
7 votes
0 answers
202 views

What (if any) was known about null sets before Lebesgue?

The notion af a null set, i. e., a set of Lebesgue measure zero, does not require a full blown construction of Lebesgue measure: A set is $E\subset \mathbb{R}$ is called a null-set if it can be ...
Kostya_I's user avatar
  • 8,632
7 votes
0 answers
245 views

When is Radon-Nikodym derivative induced by a proper map of manifolds bounded?

Let $X,Y$, be compact complex manifolds, and let $f:X\to Y$ be a smooth, proper (i.e. for each $y\in Y$, $f^{-1}(y)$ is a compact set) and surjective map. Choose metrics on $X,Y$ and let $\mu_X, \mu_Y$...
Mozhgan Mirzaei's user avatar
7 votes
0 answers
498 views

Counter-example to the completeness of the Wasserstein metric

$\newcommand{\P}{\mathcal{P}}$ Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
Oleg's user avatar
  • 911
7 votes
0 answers
452 views

Characterizing the sum $L^1 + L^\infty + L^{1,\infty} + L^{\infty, 1}$ of iterated Lebesgue spaces "by duality"

For the usual Lebesgue spaces $L^p (\mu)$ ($p \in [1,\infty]$) on a ($\sigma$-finite) measure space $(X,\mu)$, it is well-known that one has the characterization $$ L^p (\mu) = \left\{f : X \to \Bbb{...
PhoemueX's user avatar
  • 754
7 votes
0 answers
3k views

What is vague convergence and what does it accomplish?

For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
Greg Zitelli's user avatar
7 votes
0 answers
305 views

Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action

Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...
Stéphane Laurent's user avatar
7 votes
0 answers
2k views

Optimization over space of probability measures

Consider an optimization problem as follows: $$ \min\mathbb E_w[f_0(w)] \mathrm{\,\,\,\,\,\ s.t.\,\,\,\,} E_w[f_i(w)]\leq 0 ,\,\,\, i=1,\dots, k $$ where the maximum is taken over $\mathscr M$, ...
Arash's user avatar
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